The conversion of a wavelength through use of high power density within a resonator has been proposed. For example, an external resonator type second harmonics generator (SHG) has been proposed in addition to an alternate type of second harmonics generator known as internal resonator type SHG which uses a non-linear optical crystal element within a resonator having a light source. As an example of the external resonator type SHG, Japanese Laid Open Patent H5-243361 (1993) discloses a SHG using beta barium borate (BBO) as a non-linear optical crystal element. Also known is an internal resonator type SHG using potassium titanyl phosphate (KTiOPO.sub.4) or KTP as a non-linear optical crystal element. In the external or internal resonator type SHG, the laser light of the second harmonics can be efficiently output by phase-matching the laser light of the second harmonics with respect to the laser light of the fundamental frequency.
The conversion efficiency in a second harmonics generator is explained as follows:
FIG. 2 shows an apparatus for generating second harmonics by an external type resonator. A light source 11 generates a laser light having a fundamental frequency. The laser light is radiated to an external type resonator 30 through a phase modulator 12 for producing a frequency error signal. Further, the laser light is transmitted through a lens system 13 for mode matching to the external resonator 30. The external resonator 30 is made up of two concave mirrors 18 and 19 and a plane mirror 20 having design parameters as shown in Table 1.
TABLE 1 ______________________________________ Reflectance R Fundamental Transmittance T Radius of Wavelength SHG light Curvature (532 nm) (266 nm) ______________________________________ Mirror 18 50 mm 99.0% -- Mirror 19 50 mm 99.9% 90% or more Mirror 20 Flat 99.9% -- ______________________________________
An electro-magnetic actuator 16 is employed for positioning the mirror 18. A non-linear optical crystal element 17 fabricated from BBO is placed within the resonator 30. The laser light of the fundamental frequency, reflected by the resonator 30, is detected by a photodetector 14. The electro-magnetic actuator 16 is position-controlled by a control circuit 15, using a detection signal of the light detector 14. Thereby, the incident light resonates with the resonator length to efficiently produce a laser light of second harmonics by the non-linear optical crystal element 17. Japanese Laid Open Patent H5-243661 (1993) discloses that the electro-magnetic actuator is position-controlled by servo control. The Japanese patent discloses that a detection signal of a reflected light from the photodetector 14 and a modulated signal for driving the phase modulator 12 are synchronously detected and passed through a low-pass filter to produce an error signal indicating an error of an optical path length of the resonator. Then, the electromagnetic actuator 16 is driven by the error signal for moving the mirror 18 in the direction of the optical axis until the error signal becomes zero.
The SHG conversion efficiency .eta..sub.SH is found by the following Equation (1). EQU .eta..sub.SH =.gamma..sub.SH P.sub.c ( 1)
Here, .gamma..sub.SH is a non-linear conversion factor and Pc is the power incident to the resonator. A spot radius .omega..sub.0 of BBO as an example of the non-linear optical crystal element 17 is calculated to be equal to 49 .mu.m when the distance between the mirrors 16 and 20 is approximately 85 mm. Thus, for the crystal thickness of 3 mm, the non-linear conversion factor .gamma..sub.SH is found to be 1.5.times.10.sup.-5 (W.sup.-1).
The internal power in the resonator 30 during resonation is multiplied by multiple reflection. Thus, the conversion efficiency may be expected to be improved by placing a non-linear optical crystal element within the resonator 30.
The multiplication factor may be found from the Fabry-Perot's equation of multiple reflection. Thus, the amplitude reflection ratio r may be found from the following Equation (2), when R.sub.1 is the reflectance of an incident mirror 18 and R.sub.m is the reflectance of the outgoing light inclusive of the round trip loss within the resonator 30 and the reflectance of the mirrors 19 and 20. ##EQU1##
The amplitude multiplication factor tc within the resonator 30 is found from the following Equation (3). ##EQU2## Here, .DELTA. is the round trip phase difference and T.sub.1 =1-R.sub.1.
From the above Equations (2) and (3), the intensity reflection ratio R my be found by the following Equation (4) ##EQU3## The intensity multiplication factor Tc is found by the following Equation (5). ##EQU4##
For R.sub.1 =R.sub.m and R.sub.2 =1, that is when the total loss including the loss by SHG within the resonator is equal to the incident transmittance, the reflectance during resonation becomes zero, such that all the energy of the incident light is input to the resonator. This state is termed an impedance-matched state. Since R.sub.1 =R.sub.m, R.sub.2 =1 and T.sub.1 =1-R.sub.m for .DELTA.=2 m.pi., the intensity multiplication factor T.sub.c during the impedance-matched state is found from the following Equation (6). ##EQU5## Here, P.sub.i is the incident power and P.sub.c is the power within the resonator.
Substituting Rm=99%, it is found that the 100 times multiplication effect may be achieved. Next, the SHG conversion efficiency in this state is scrutinized. If the SHG conversion efficiency is regarded as being an increase in the loss within the resonator, it is found from Equation (7). EQU R.sub.m =1-.delta..sub.cav -.eta..sub.SH ( 7)
Here, R.sub.m is the reflectance at the outgoing light side (towards rear), .delta..sub.cav is the loss within the resonator and .eta..sub.SH is the SHG conversion efficiency for a single pass.
Substituting the Equation (7) into the Equation (6), a quadratic Equation (8) concerning the internal power P.sub.c ##EQU6## is obtained.
The power within the resonator P.sub.c is found from the following Equation (9). ##EQU7##
Thus, the effective SHG conversion efficiency .eta. for the incident power may be found from the following Equation (10). ##EQU8## Here, X is represented by the following Equation (11). ##EQU9##
If the non-linear optical crystal element 17 is fabricated from BBO as an example, the crystal thickness is 3 mm and the spot radius is 49 .mu.m, .gamma..sub.SH =1.5.times.10.sup.-5 (W.sup.-1). Substituting the input power P.sub.i =1 W and the resonator loss .delta..sub.cav =0.5% into the Equations (11) and (12), it may be seen that the conversion efficiency amounts to about 30%. Unless impedance matching is achieved, the conversion efficiency may be found as the numerical solution by substituting the above Equation (7) into the Equation (5).
Although the conversion efficiency of approximately 30% represents significant improvement over that of the single-pass SHG, the remaining 70% represents heat which is wastefully consumed. In addition, there is raised a problem that the conversion efficiency is changed significantly with changes in the value of the resonator losses.
If the Equation (10) is rewritten using the Equations (1) and (8), the Equation (12) is obtained. ##EQU10##
Since the resonator loss is the diffusion and absorption by the mirror or the non-linear optical crystal element, it is impossible to reduce the resonator loss to zero. Consequently, as long as the resonator loss exists, it is impossible to get 100% conversion efficiency. In addition, if the value of the resonator loss is changed, the conversion efficiency is changed significantly. The SHG conversion efficiency within the SHG of the internal type resonator is similar to that of the SHG of the external type resonator.
The above-described SHG laser source of the external resonation type or the SHG laser light source by the non-linear optical crystal element within the laser resonator (the SHG of the internal resonator type) have several disadvantages. One disadvantage is that resonator loss results in less than 100% conversion. A second disadvantage is that the conversion efficiency is significantly changed by fluctuations in resonator loss.